In article <cs5bce$3tk$1 at smc.vnet.net>,
Ken Tozier <kentozier at comcast.net> wrote:
> I'm self taught so I wasn't aware of the official name of the class of
> curves I'm looking at, but have since found that they are called
> "trochoids" and they have a whole section devoted to them here:
> http://mathworld.wolfram.com/Trochoid.html. Integrals might as well be
> written in Martian for all the meaning I get out of them. I find them
> impenetrable, so I don't know if the trochoids arc length formula
> here: http://mathworld.wolfram.com/CurtateCycloid.html is considered
> "closed form" or not.
Equation (3) at this page is a closed form expression for the arc length
and Mathematica can compute the complete elliptic integral of the second
kind and Jacobi elliptic functions to arbitrary precision. However, it
is not obvious to me how the sum you wrote down, essentially
Sum[Sqrt[ (d^2 + 2 s^2 - 2 s (s Cos[(2 Pi)/s] +
d (Cos[(2 k Pi)/s] - Cos[(2 (1 + k) Pi)/s])))]/s,
{k, 0, Infinity}]
relates to the arc length. I assume this is some sort of Riemann sum
approximation to the integral? Note that there is a Mathematica Notebook
at http://mathworld.wolfram.com/CurtateCycloid.html and in that Notebook the
arc length is computed in closed form using Mathematica as
2 (a - b) EllipticE[t/2, - 4 a b/(a - b)^2]
for a > b.
Cheers,
Paul
--
Paul Abbott Phone: +61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul